Stefan Becuwe

Adaptive multivariate rational data fitting with applications in electromagnetics

The behavior of certain electromagnetic devices or components can be simulated with great detail in software. A drawback of these simulation models is that they are very time consuming. Since the accuracy required for the computational electromagnetic analysis is usually only 2-3 significant digits, an approximate analytic model is sometimes used instead, as noted by Lehmensiek and Meyer in 2001. The most complex model we consider here is a multivariate rational function, which interpolates a number of simulation data. The interpolating rational function is constructed in such a way that it minimizes both the truncation error and the number of simulation data since each evaluation of the simulation model is computationally costly.

A Remarkable Example of Catastrophic Cancellation Unraveled

Abstract In this paper we reinvestigate a well-known expression first published in [7], which is often used to illustrate catastrophic cancellation as well as the fact that identical output in different precisions does not imply reliability. The purpose of revisiting this expression is twofold. First, we show in Section 2 that the effect of the cancellation is very different on different IEEE 754 compliant platforms, and we unravel the underlying (hardware) reasons which are unknown to many numerical analysts. Besides illustrating cancellation, this expression also counters the common misbelief among many numerical analysts that a same program will deliver identical results on all IEEE conforming systems. Second, in Section 3 we use, illustrate and comment upon the cross-platform didactical tool Arithmetic Explorer developed at the University of Antwerp, by means of which we performed the bit level analysis of the expression evaluation under investigation on the different machines. We believe that this tool, which is freely available from the authors, can be of use to all of us teaching a first numerical analysis course.

Validated Evaluation of Special Mathematical Functions

Because of the importance of special functions, several books and a large collection of papers have been devoted to the numerical computation of these functions, the most well-known being the Abramowitz and Stegun handbook [1]. But up to this date, no environment offers routines for the provable correct evaluation of these special functions. We point out how series and limit-periodic continued fraction representations of the functions can be helpful in this respect. Our scalable precision technique is mainly based on the use of sharpened a priori truncation and round-off error upper bounds, in case of real arguments. The implementation is validated in the sense that it returns a sharp interval enclosure for the requested function evaluation, at the same cost as the evaluation.

Multivariate Rational Interpolation of Scattered Data

Rational data fitting has proved extremely useful in a number of scientific applications. We refer among others to its use in some network problems [6,7,15,16], to the modelling of electro-magnetic components [20,13], to model reduction of linear shift-invariant systems [2,3,8] and so on.

On the Froissart phenomenon in multivariate homogeneous Padé approximation

In univariate Padé approximation we learn from the Froissart phenomenon that Padé approximants to perturbed Taylor series exhibit almost cancelling pole–zero combinations that are unwanted. The location of these pole–zero doublets was recently characterized for rational functions by the so‐called Froissart polynomial. In this paper the occurrence of the Froissart phenomenon is explored for the first time in a multivariate setting. Several obvious questions arise. Which definition of Padé approximant is to be used? Which multivariate rational functions should be investigated? When considering univariate projections of these functions, our analysis confirms the univariate results obtained so far in [13], under the condition that the noise is added after projection. At the same time, it is apparent from section 4 that for the unprojected multivariate Froissart polynomial no conjecture can be formulated yet.

Validated evaluation of special mathematical functions

Because of the importance of special functions, several books and a large collection of papers have been devoted to their use and computation, the most well-known being the Abramowitz and Stegun handbook (Abramowitz and Stegun, 1964) 1] and its successor (Olver et al. 0000) 2]. However, until now no environment offers routines for the provable correct multiprecision and radix-independent evaluation of these special functions. We point out how we make good use of series and limit-periodic continued fraction representations in a package that is being developed at the University of Antwerp. Our scalable precision technique is mainly based on the use of sharpened a priori truncation and round-off error upper bounds for real arguments. The implementation is validated in the sense that it returns a sharp interval enclosure for the requested function evaluation, at the same cost as the evaluation.

Towards reliable software for the evaluation of a class of special functions

Special functions are pervasive in all fields of science. The most well-known application areas are in physics, engineering, chemistry, computer science and statistics. Because of their importance, several books and a large collection of papers have been devoted to the numerical computation of these functions. But up to this date, even environments such as Maple, Mathematica, MATLAB and libraries such as IMSL, CERN and NAG offer no routines for the reliable evaluation of special functions. Here the notion reliable indicates that, together with the function evaluation a guaranteed upper bound on the total error or, equivalently, an enclosure for the exact result, is computed. We point out how limit-periodic continued fraction representations of these functions can be helpful in this respect. The newly developed (and implemented) scalable precision technique is mainly based on the use of sharpened a priori truncation error and round-off error upper bounds for real continued fraction representations of special functions of a real variable. The implementation is reliable in the sense that it returns a sharp interval enclosure for the requested function evaluation, at the same cost as the evaluation.

Validated special functions software

Because of the importance of special functions, several books and a large collection of papers have been devoted to the numerical computation of these functions, the most well-known being the NBS handbook by Abramowitz and Stegun. But up to this date, symbolic and numeric environments offer no routines for the validated evaluation of special functions. We point out how a provable correct function evaluation can be returned efficiently.

Continued Fractions for Special Functions: Handbook and Software

The revived interest in continued fractions stems from the fact that many special functions enjoy easy to handle and rapidly converging continued fraction representations. These can be made to good use in a project that envisages the provably correct (or interval) evaluation of these functions. Of course, first a catalogue of these continued fraction representations needs to be put together. The Handbook of continued fractions for special functions is the result of a systematic study of series and continued fraction representations for several families of mathematical functions used in science and engineering. Only 10% of the listed continued fraction representations can also be found in the famous NBS Handbook edited by Abramowitz and Stegun. More information is given in Sect. 1. The new handbook is brought to life at the website www.cfhblive.ua.ac.be where visitors can recreate tables to their own specifications, and can explore the numerical behaviour of the series and continued fraction representations. An easy web interface supporting these features is discussed in the Sects. 2, 3 and 4.